The invention relates to estimating liquid flow rates in open-channel geometries and relates particularly, though not exclusively, to an improved method of effectively estimating liquid fuel flow rates and other parameters associated with the flow of liquid fuel through vane-type propellant management devices under zero-gravity conditions, that is, in fuel delivery systems used by space equipment.
In vane-type surface tension propellant management devices (PMD) commonly used in satellite fuel tanks, the propellant is transported along guiding vanes from a reservoir at the inlet of the device to a sump at the outlet from where the fuel is pumped to the satellite engine. The pressure gradient driving this free-surface flow under zero-gravity (zero-g) conditions is generated by surface tension and is related to the differential curvatures of the propellant-gas interface at the inlet and outlet of the PMD.
Constructional details of exemplary designs and systems can be found in the existing patent literature. In this respect, by way of example only, U.S. Pat. Nos. 4,733,531, 5,293,895, 4,553,565 and 4,743,278 disclose typical constructions.
In a real-life situation, after one set of satellite orbiting manoeuvres are completed, the designer/operator would like to know for how long one has to wait before the sump gets refilled, so that the next set of manoeuvres may be performed. Given the complexity of the governing equations in this free-surface flow problem, the only possible options are to generate the required data for the drainage times either experimentally or via numerical simulation. If one decides to proceed experimentally, the options are to either perform the experiment in space (which would obviously be expensive) or on the ground, where one would be faced with the cost and difficulties of setting up a zero-g environment.
Experimental approaches involve obvious difficulties and limitations, a notable one of which is expense. An overview of some of the other issues involved in using experimental techniques are discussed (in the context of space shuttle fuel tanks) in Dale A. Fester, Ashton J. Villars, and Preston E. Uney, Surface tension propellant acquisition system technology for space shuttle reaction control tanks. J. Spacecraft 9, 522 (1976).
The second option, of using computer simulation, presents equally difficult challenges. One computes an unsteady, three-dimensional free-surface flow in a complex geometry. Further, the flow rates involved in a zero-g environment are extremely small; typically, the drainage time for 2 litres of propellant could be anywhere from 8 to 24 hours. These flow rates are several orders of magnitude less than those in a 1-g environment.
Computing these zero-g flows by a brute-force unsteady 3-D simulation would not only be prohibitively expensive, but the issues of convergence and accuracy would be difficult to settle because of the extremely small flow rates involved. If one attempts to take care of the complexity of the geometry by ignoring the entry and exit regions, then the problem would be to accurately estimate the entry and exit conditions in a straight section of the vane.
To the inventor""s knowledge there do not currently exist any published numerical simulation results for a zero-g PMD. Estimates of drainage times, presumably obtained experimentally, are available in several references, typically papers of the American Institute of Aeronautics and Astronautics, such as: (a) A. Kerebel and D. Baralle, A low-cost surface tension tank optimised for telecommunication satellites. AIAA-85-1131; (b) A. Kerebel and P. Durgat, Development of a telecommunications spacecraft propellant tank. AIAA-86-1502; or (c) D. Baralle and J. P. Fournier, Propellant tank for telecommunication platforms. AIAA-89-2761.
Owing to the substantial cost and the technical hurdles involved in accurately estimating these minuscule flow rates by either direct numerical simulation or by experimental methods which simulate zero-g conditions in the lab, any solution which offers advantages of any kind over existing techniques would provide benefits to those working in the field, particularly in relation to the design and general operation of satellite fuel tanks.
In view of the above, a need clearly exists for an improved method of calculating flow rates in PMD devices that at least attempts to address one or more of the limitations of existing techniques.
The aspects of the invention involve a recognition that an accurate and computationally tractable solution to the problem of calculating the fuel flow rates in a zero-g PMD can be achieved by using a semi-analytical procedure under certain reasonably idealized conditions. As the inventive technique uses exact analytical solutions (via eigenfunction expansions) of a suitably perturbed version of the Stokes flow equations, the issue of convergence is confined to the accuracy with which boundary data are satisfied. Further, due to relatively conservative computational and memory requirements, the effect of varying parameters (such as, for example, aspect ratio and contact angle) can be more readily investigated.
Accordingly, a first aspect of the invention provides a method of estimating the flow rate of a liquid through an open-channel geometry, the method comprising steps of:
defining (i) an open-channel geometry having an inlet end and an outlet end, between which there is a capillary passage having two parallel walls, (ii) a set of parameters associated with the open-channel geometry, and (iii) analytical equations involving the set of parameters which govern the flow of liquid through the open-channel geometry; and
deriving a modified set of analytic equations corresponding to the governing analytic equations of the open-channel geometry, and solving said modified set of analytic equations to calculate a representative flow rate defined by the modified set of equations;
wherein said modified set of equations are derived from a modified open-channel geometry based on an artificial assumption that an elongate dimension of the open-channel geometry is tapered, so that the representative flow rate can be calculated as an approximation of the flow rate in the open-channel geometry.
In the modified set of equations, the taper of the channel geometry is assumed to be linear in the height of the open-channel geometry. The representative flow rate is calculated on the assumption that the flow rate is independent of the geometries of the inlet and outlet ends. Further, time derivatives are assumed to be negligible so that the flow rate is directly related to the instantaneous pressure gradient between the inlet end and the outlet end.
The artificial assumption of a tapered channel height obviates the need to take into account the calculation of wall layers in the open-channel geometry for the analytic governing equations, so that the free surface of the liquid has approximately a constant radius of curvature which is able to satisfy requirements of a defined contact angle between the liquid and the open-channel geometry.
The calculated meniscus radii of curvature of the liquid at the inlet and outlet ends of the open-channel geometry is assumed to be independent of the inlet or outlet end geometry, in the modified set of equations. The instantaneous pressure difference along the open-channel geometry is determined from the Laplace-Young equation and from the respective inlet and outlet meniscus radii and the surface tension coefficient of the liquid.
A parameter xcex4 defines a linear taper in average channel height H, such that the inlet and outlet heights Hi and Ho are given by:             H      i        =                  2        ⁢        H                    1        +        δ              ,      xe2x80x83    ⁢            H      o        =                            2          ⁢                      xe2x80x83                    ⁢          δ          ⁢                      xe2x80x83                    ⁢          H                          1          +          δ                    =              δ        ⁢                  xe2x80x83                ⁢                              H            i                    .                    
In the modified equations, the pressure difference for the open-channel geometry is equated with the pressure difference for the modified open-channel geometry to determine the parameter xcex4.
The modified set of equations are preferably solved with a finite difference scheme using a weighted least-squares algorithm.
The parallel walls of the open-channel geometry represent the fuel tank wall and a vane in a propellant management device, with the other two sides being open.
The invention also provides a computer program product having a computer readable medium having a computer program recorded therein for estimating the flow rate of a liquid through an open-channel geometry, said computer program comprising:
means for recording a definition of (i) an open-channel geometry having an inlet end and an outlet end, between which there is a capillary passage having two parallel walls, (ii) a set of parameters associated with the open-channel geometry, and (iii) analytical equations involving the set of parameters which govern the flow of liquid through the open-channel geometry; and
means for storing a modified set of analytic equations corresponding to the governing analytic equations of the open-channel geometry, and solving said modified set of analytic equations to calculate a representative flow rate defined by the modified set of equations;
wherein said modified set of equations are derived from a modified open-channel geometry based on an artificial assumption that an elongate dimension of the open-channel geometry is tapered, so that the representative flow rate can be calculated as an approximation of the flow rate in the open-channel geometry.
The computer program is suitable for execution on existing general-purpose computing hardware.